The absolute value of the hyperdeterminant can be used as a multi-partite entanglement measure [1]}, known as the concurrence [2]}, [3]} and three-tangle [4]} for two- and three-qubit pure states respectively. For general two- and three-qubit states, expressed as \(|\Psi ^{(2)}\rangle \equiv \sum _{i,j=1,2}\Gamma _{ij}|ij\rangle \)  and \(|\Psi ^{(3)}\rangle \equiv \sum _{i,j,k=1,2}\Gamma _{ijk}|ijk\rangle \) , the concurrence and the three-tangle can be explicitly written as \(C(|\Psi ^{(2)}\rangle )\equiv 2|\det (\mathbf {\Gamma }^{(2)})|=2|\Gamma _{00}\Gamma _{11}-\Gamma _{01}\Gamma _{10}|\)  and \(\tau _3(|\Psi ^{(3)}\rangle )\equiv 4|\operatorname{Det}(\mathbf {\Gamma }^{(3)})|\)  respectively [1]}, where \(\mathbf {\Gamma }^{(2)}\equiv [\Gamma _{ij}]\)  is a matrix, \(\mathbf {\Gamma }^{(3)}\equiv [\Gamma _{ijk}]\)  is a third-order tensor, and \(\operatorname{Det}(\mathbf {\Gamma }^{(3)})\)  is Cayley's hyperdeterminant defined by [1]}, [7]}
\(\operatorname{Det}(\mathbf {\Gamma }^{(3)})&=\left(\begin{vmatrix}\Gamma _{000} & \Gamma _{011}\\\Gamma _{100} & \Gamma _{111}\end{vmatrix}+\begin{vmatrix}\Gamma _{010} & \Gamma _{001}\\\Gamma _{110} & \Gamma _{101}\end{vmatrix}\right)^2\nonumber \\&-4 \begin{vmatrix}\Gamma _{000} & \Gamma _{001}\\\Gamma _{100} & \Gamma _{101}\end{vmatrix}\cdot \begin{vmatrix}\Gamma _{010} & \Gamma _{011}\\\Gamma _{110} & \Gamma _{111}\end{vmatrix}.\) 
